A pyramid has a square base and a volume of 144 cubic centimeters. The area of the base of the...

1. TRANSLATE the problem information

• Given information:
  • Pyramid has a square base
  • Volume = 144 cubic centimeters
  • Base area = 36 square centimeters
• Need to find: height of the pyramid

2. INFER the appropriate formula

• Since we know volume and base area, and need height, we should use the pyramid volume formula
• The volume formula connects all three quantities: \(\mathrm{V = \frac{1}{3} \times base\ area \times height}\)

3. TRANSLATE the given values into the formula

• Substitute: \(\mathrm{144 = \frac{1}{3} \times 36 \times h}\)
• This gives us: \(\mathrm{144 = 12h}\)

4. SIMPLIFY to solve for height

• Divide both sides by 12: \(\mathrm{h = 144 \div 12}\)
• Calculate: \(\mathrm{h = 12}\) centimeters

Answer: B. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about pyramid volume formula: Students might confuse the pyramid volume formula with other volume formulas (like rectangular prism V = length × width × height) and forget the \(\mathrm{\frac{1}{3}}\) factor.

Without the \(\mathrm{\frac{1}{3}}\) factor, they would calculate: \(\mathrm{144 = 36h}\), leading to \(\mathrm{h = 4}\). This may lead them to select Choice A (4).

Second Most Common Error:

Weak SIMPLIFY skills: Students might make arithmetic errors when solving the equation, particularly when dealing with the fraction \(\mathrm{\frac{1}{3}}\) or the final division step.

Some might incorrectly multiply instead of divide, calculating \(\mathrm{h = 144 \times 12 = 1,728}\), or make other computational mistakes. This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students remember the specific volume formula for pyramids (with the 1/3 factor) and can execute basic algebraic manipulation accurately. The pyramid volume formula is less commonly used than other geometric formulas, making it a frequent source of errors.